You almost certainly know what a proportion is in math; it’s just an equation with a ratio (expressed as a fraction) on each side of the equals sign. Proportions are commonly used to represent a relationship between two quantities, as often is necessary on SAT math problems. For example, we can express the relationship between feet and yards with this proportion:
As with all equations, it doesn’t matter which expression is on which side of the equals sign; we could also have written
It also doesn’t matter which way we orient the ratios with respect to the numerators and denominators, as long as we keep the two sides correlated, so we could also write,
A less appreciated fact is that you can turn a proportion “sideways”:
A more realistic example of a proportion that comes up on the SAT involves the conversion between degrees and radians. Here’s one of the ways we can express that relationship:
If you’re one of those semicircle people who are into reducing fractions, you can, of course, express this as
(If you don’t get it, ask your parents, though they might not get it, either)
To use a proportion to solve for an unknown, we’d typically have three out of the four values, with the fourth value unknown. For example, we might know that a given central angle measures 160°, and we’d like to know what that is in radians, for which we’ll use ; we can use our proportion to say
To solve for , we can use the usual algebraic manipulations; we’ll start by getting in a numerator by multiplying both sides by :
Next, we will eliminate the denominator on the right side by multiplying both sides by :
Finally, we will divide both sides by 360 to isolate :
At this point, we can perform the chain calculation on the left side without needing to store any intermediate results:
Now, we have our answer; 160° is equivalent to about 2.79 radians.
You might think it doesn’t matter how you set up your proportion, because any valid arrangement will express the same relationship. However, that’s not the case, and we at 1600.io have a recommendation for you: place the unknown in the numerator of the ratio on the left side of the equation.
Let’s work through the previous example again, but using this recommended setup.
Notice that the unknown is now in the numerator of the fraction on the left side. As our first step, we’ll multiply both sides by 160 to isolate :
Notice anything? Yep — we’re done! All we have to do is perform the chain calculation on the right side, which can be entered into a calculator with no parentheses or storing of intermediate results, and we have our answer, expressed in a sensible statement:
That’s the beauty of setting up the proportion with the unknown in the numerator. And why do we recommend putting it in the numerator on the left side, if an equation is symmetrical? Only because the final result is a more satisfying mathematical “sentence,” as it will read from left-to-right as “ equals 2.79.”
